A growing number of researchers have recently published techniques that perform blind source separation (BSS), i.e., separating a composite signal into its constituent component signals without a priori knowledge of those signals. These techniques find use in various applications such as speech detection using multiple microphones, crosstalk removal in multichannel communications, multipath channel identification and equalization, direction of arrival (DOA) estimation in sensor arrays, improvement of beam forming microphones for audio and passive sonar, and discovery of independent source signals in various biological signals, such as EEG, MEG and the like. Many of the BSS techniques require (or assume) a statistical dependence between the component signals to accurately separate the signals. Additional theoretical progress in signal modeling has generated new techniques that address the problem of identifying statistically independent signals--a problem that lies at the heart of source separation.
The basic source separation problem is simply described by assuming d.sub.s statistically independent sources s(t)=[s.sub.1 (t), . . . , s.sub.d.sbsb.s (t)].sup.T that have been convolved and mixed in a linear medium leading to d.sub.x sensor signals x(t)=[x.sub.1 (t), . . . , x.sub.d.sbsb.s (t)].sup.T that may include additional sensor noise n(t). The convolved, noisy signal is represented in the time domain by the following equation (known as a forward model): ##EQU1## Source separation techniques are used to identify the d.sub.x d.sub.s P coefficients of the channels A and to ultimately determine an estimate s(t) for the unknown source signals.
Alternatively, the convolved signal may be filtered using a finite impulse response (FIR) inverse model represented by the following equation (known as the backward model): ##EQU2## In this representation, a BSS technique must estimate the FIR inverse components W such that the model source signals u(t)=[u.sub.1 (t), . . . , u.sub.d.sbsb.u (t)].sup.T are statistically independent.
An approach to performing source separation under the statistical independence condition has been discussed in Weinstein et al., "MultiChannel Signal Separation by Decorrelation", IEEE Transaction on Speech and Audio Processing, vol. 1, no. 4, pp. 405-413, 1993, where, for non-stationary signals, a set of second order conditions are specified that uniquely determine the parameters A in the forward model. However, no specific algorithm for performing source separation based on non-stationarity is given in the Weinstein et al. paper.
Early work in the signal processing community had suggested decorrelating the measured signals, i.e., diagonalizing measured correlations for multiple time delays. For an instantaneous mix, also referred to as the constant gain case, it has been shown that for non-white signal decorrelation using multiple filter taps is sufficient to recover the source signals. However, for convolutive mixtures of wide-band signals, this technique does not produce a unique solution and, in fact, may generate source estimates that are decorrelated but not statistically independent. As clearly identified by Weinstein et al. in the paper cited above, additional conditions are required to achieve a unique solution of statistically independent sources. In order to find statistically independent source signals, it is necessary to capture more than second order statistics, since statistical independence requires that not only second but all higher cross moments vanish.
In the convolutive case, Yellin and Weinstein in "Multichannel Signal Separation: Methods and Analysis", IEEE Transaction on Signal Processing; vol. 44, no. 1, pp. 106-118, January 1996 established conditions on higher order multi-tap cross moments that allow convolutive cross talk removal. Although the optimization criteria extends naturally to higher dimensions, previous research has concentrated on a two dimensional case because a multi-channel FIR model (see equation 2) can be inverted with a properly chosen architecture using estimated forward filters. Heretofore, for higher dimensions, finding a stable approximation of the forward model has been illusive.
These prior art techniques generally operate satisfactorily in computer simulations but perform poorly for real signals, e.g., audio signals. One could speculate that the signal densities of the real signals may not have the hypothesized structures, the higher order statistics may lead to estimation instabilities, or a violation of the signal stationarity condition may cause inaccurate solutions.
Therefore, there is a need in the art for a blind source separation technique that accurately performs convolutive signal decorrelation.